Bartoli–Timpanella conjecture on a power function with planar-like differential uniformity

Let $p$ be an odd prime and $n$ an odd integer. For a function $F$ over ${\mathrm{GF}}(p^n)$, write that $F$ is P$c$N for $c=-1$ when its $c$-differential uniformity is $1$. Consider the power function

$$F(x)=x^{\frac{p^n+1}{p+1}}.$$

Bartoli–Timpanella conjecture. For $c=-1$, the power function $F$ is P$c$N over ${\mathrm{GF}}(p^n)$. This conjecture concerns the existence of power functions with perfect $c$-nonlinearity, a generalized form of planarity, and was proposed by Bartoli and Timpanella; its resolution status is not specified in the source.